How do you write an equation for the hyperbola with center at (-2,-3), Focus at (-4,-3), Vertex at (-3,-3)?

Feb 13, 2016

Equation of hyperbola is ${\left(x + 2\right)}^{2} / 1 - {\left(y + 3\right)}^{2} / 3 = 1$

Explanation:

As $y$ coordinates of center, focus, and vertex all are $- 3$, they lie on the horizontal line $y = - 3$ and general form of such hyperbola is

${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$, where$\left(h , k\right)$ is center.

Here center is $\left(- 2 , - 3\right)$.

Further, $a$ is distance of vertex from center and ${a}^{2} + {b}^{2} = {c}^{2}$, where $c$ is distance of focus from center.

As the vertex is 1 units from the center, so $a = 1$; the focus is 2 units from the center, so $c = 2$.

As ${a}^{2} + {b}^{2} = {c}^{2}$, ${b}^{2} = {c}^{2} - {a}^{2} = 4 - 1 = 3$.

Equation of hyperbola is hence

${\left(x + 2\right)}^{2} / 1 - {\left(y + 3\right)}^{2} / 3 = 1$